Rotating around two z axes
You'll probably want to use an extra empty object that you rotate the moon around. That way you can rotate the earth and the moon independently from each other. Before starting, reset the origins of your objects (ShiftCtrlAltC).

To do so:


Start by adding an empty object in the same location as your earth object (you can snap your cursor to the earth by using ShiftS > Cursor to Selected)
Select your moon object, then select your empty and parent it (CtrlP). You will notice that you can now rotate the moon with the empty, but the earth rotates separately.
Rotate the earth at the speed you want.
Rotate the moon.
To make the moon orbit around the earth, rotate the empty.


If you want to make the moon orbit copy the speed of the earth, you can use a copy rotation constraint on the empty (you can also set the percentage of the speed by adjusting the influence).